tilburg university investment under vanishing uncertainty …no. 2001-14 investment under vanishing...

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Tilburg University Investment Under Vanishing Uncertainty Due to Information Arriving Over Time Thijssen, J.J.J.; van Damme, E.E.C.; Huisman, K.J.M.; Kort, P.M. Publication date: 2001 Document Version Publisher's PDF, also known as Version of record Link to publication in Tilburg University Research Portal Citation for published version (APA): Thijssen, J. J. J., van Damme, E. E. C., Huisman, K. J. M., & Kort, P. M. (2001). Investment Under Vanishing Uncertainty Due to Information Arriving Over Time. (CentER Discussion Paper; Vol. 2001-14). Microeconomics. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 26. Aug. 2021

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Page 1: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

Tilburg University

Investment Under Vanishing Uncertainty Due to Information Arriving Over Time

Thijssen, J.J.J.; van Damme, E.E.C.; Huisman, K.J.M.; Kort, P.M.

Publication date:2001

Document VersionPublisher's PDF, also known as Version of record

Link to publication in Tilburg University Research Portal

Citation for published version (APA):Thijssen, J. J. J., van Damme, E. E. C., Huisman, K. J. M., & Kort, P. M. (2001). Investment Under VanishingUncertainty Due to Information Arriving Over Time. (CentER Discussion Paper; Vol. 2001-14). Microeconomics.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 26. Aug. 2021

Page 2: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

No. 2001-14

INVESTMENT UNDER VANISHING UNCERTAINTYDUE TO INFORMATION ARRIVING OVER TIME

By Jacco J.J. Thijssen, Eric E.C. van Damme, Kuno J.M. Huisman and Peter M. Kort

February 2001

ISSN 0924-7815

Page 3: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

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Page 4: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

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Page 5: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

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Page 6: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

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Page 7: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

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Page 8: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

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Xqghu wkh qhw suhvhqw ydoxh +QSY, dssurdfk/ wkh prqrsrolvw lv lqgl�huhqw

ehwzhhq lqyhvwlqj dqg qrw lqyhvwlqj li LH+Xmq> j, htxdov }hur1 Li lw lv vpdoohu

wkdq }hur kh zloo devwdlq iurp lqyhvwlqj dqg li lw lv odujhu wkdq }hur kh zloo

lqyhvw lpphgldwho|1 Khqfh/ wkhuh lv d fulwlfdo ohyho iru s+q> j,/ zklfk zh

ghqrwh e| s��T / dw zklfk wkh prqrsrolvw lv lqgl�huhqw1 Wklv fulwlfdo ohyho

9

Page 9: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

lv jlyhq e|/

s��T @L

XM= +:,

Li wkh prqrsrolvw eholhyhv wkdw/ jlyhq wkh qxpehu ri vljqdov dqg wkh qxpehu

ri jrrg vljqdov/ wkh suredelolw| wkdw uhyhqxhv duh kljk htxdov s��T / wkhq/

xqghu wkh QSY dssurdfk/ kh lv lqgl�huhqw ehwzhhq lqyhvwlqj dqg qrw lqyhvw0

lqj1 Wklv lpsolhv wkdw wkhuh lv d fulwlfdo ohyho ri frpelqdwlrqv ri q dqg j

wkdw pdnh wkh prqrsrolvw lqgl�huhqw1 Lq rwkhu zrugv/ wkh prqrsrolvw qhhgv

hqrxjk jrrg vljqdov uhodwlyh wr wkh qxpehu ri edg vljqdov wr lqyhvw lq wkh

surmhfw1

Frqfhuqlqj wkh fulwlfdo ohyho s��T wkhuh duh wzr srvvlelolwlhv1 Hlwkhu

s��T A 4 ru s��T � 416 Lq wkh iruphu fdvh/ lw lv qhyhu rswlpdo iru wkh

prqrsrolvw wr lqyhvw1 Wkh odwwhu fdvh krogv li

XM� L +;,

Wklv lv d vwudljkwiruzdug frqglwlrq/ iru li lw zhuh wkh fdvh wkdw wkh prqrs0

rolvw zrxog qrw lqyhvw hyhq li kh nqhz wkdw uhyhqxhv duh kljk iru vxuh/ wkhq

lw lv dovr qhyhu rswlpdo wr lqyhvw li wkhuh lv xqfhuwdlqw| derxw wkh pdunhw

frqglwlrqv1

��� � � ������ ���������� �����

Dv vwdwhg lq wkh lqwurgxfwlrq/ wkh xqfhuwdlqw| derxw wkh pdunhw frqglwlrqv

dqg wkh luuhyhuvlelolw| ri wkh hqwudqfh wr wkh pdunhw lqgxfh dq rswlrq ydoxh

ri zdlwlqj iru pruh vljqdov1 Lq wklv vxevhfwlrq zh zloo vkrz krz wr �qg wkh

fulwlfdo ohyho iru s+q> j, dw zklfk wkh �up lv lqgl�huhqw ehwzhhq lqyhvwlqj dqg

zdlwlqj/ zkloh wdnlqj lqwr dffrxqw wkh rswlrq ydoxh ri zdlwlqj1

Iluvw/ zh h{solflwo| fdofxodwh s+q> j,1 Wr vlpsoli| pdwwhuv frqvlghudeo|/

6Qrwh iurp ht1 +:, wkdw lw dozd|v krogv wkdw R��T D f1

:

Page 10: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

gh�qh n =@ 5j � q dqg � =@ �3RfRf

1 Zh qrz rewdlq xvlqj Ed|hv* uxoh/

s+q> j, @LS+q> jmK,LS+K,

LS+q> jmK,LS+K, . LS+q> jmO,LS+O,

@�}+4� �,?3}sf

�}+4� �,?3}sf . +4� �,}�?3}+4� sf,

@�2}3?

sf

�2}3?sf . +4� �,2}3?+4� sf,

@�&

�& . �+4� �,&� s+n,=

+<,

Wkh fulwlfdo ohyho ri n zkhuh wkh prqrsrolvw lv lqgl�huhqw ehwzhhq lqyhvwlqj

dqg qrw lqyhvwlqj lq wkh surmhfw lv ghqrwhg e| nW1 Qrwh wkdw dw dq| duulydo ri

dq k0vljqdo n lqfuhdvhv zlwk rqh/ dqg dw dq| duulydo ri dq o0vljqdo n ghfuhdvhv

zlwk rqh1 Khqfh/ hqrxjk k0vljqdov pxvw duulyh wr uhdfk wkh fulwlfdo ohyho1

Wkh fulwlfdo ohyho ri wkh frqglwlrqdo eholhi ri kljk uhyhqxhv lv ghqrwhg e|

sW @ s+nW,1

Vxssrvh wkdw wkh vwdwh ri wkh surfhvv dw d sduwlfxodu srlqw lq wlph lv

jlyhq e| n1 Wkhq wkhuh duh wkuhh srvvlelolwlhv1 Iluvw/ n pljkw eh vxfk wkdw

n � nW dqg s+n, � s

W1 Wkhq lw lv rswlpdo iru wkh prqrsrolvw wr gluhfwo|

lqyhvw lq wkh surmhfw17 Lq wklv fdvh wkh ydoxh iru wkh prqrsrolvw/ ghqrwhg e|

/ htxdov wkh h{shfwhg sd|r� jlyhq lq ht1 +9,=

+n, @ XMs+n,� L= +43,

D vhfrqg srvvlelolw| lv wkdw/ hyhq diwhu d qhz k0vljqdo duulylqj/ lw lv vwloo

qrw rswlpdo wr lqyhvw/ l1h1 n ? nW

� 41 Wkh ydoxh ri wkh rssruwxqlw| wr

lqyhvw iru wkh prqrsrolvw/ ghqrwhg e| Y�/ fdq wkhq eh frqvwuxfwhg iurp wkh

iroorzlqj vhfrqg rughu olqhdu gl�huhqfh htxdwlrq/

Y�+n, @ h3o_|

i+4� �gw,Y�+n, . �gw^s+n,+�Y�+n . 4, . +4� �,Y�+n � 4,, .

. +4� s+n,,+�Y�+n � 4, . +4� �,Y�+n . 4,,`j

@ +4� ugw,+4� �gw,Y�+n, . +4� ugw,�gw^+5s+n,�. 4� �� s+n,,

Y�+n . 4, . +s+n, . �� 5s+n,�,Y�+n � 4,` . r+gw,

/ +u . �,gwY�+n, @ �gw^+5s+n,�. 4� �� s+n,,Y�+n . 4, .

. +s+n, . �� 5s+n,�,Y�+n � 4,` . r+gw,=

+44,

7Wklv kdsshqv li &W &f ' f1

;

Page 11: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

Ht1 +44, vwdwhv wkdw wkh ydoxh ri wkh rswlrq wr hqwhu dw vwdwh n pxvw htxdo

wkh glvfrxqwhg h{shfwhg ydoxh d vpdoo dprxqw ri wlph odwhu1 Lw lv qrwklqj

hovh wkdq xvlqj wkh Ehoopdq sulqflsoh iru wkh frqwlqxdwlrq uhjlrq81 Xvlqj

ht1 +<, lw krogv wkdw

5s+n,�. 4� �� s+n, @�&n� . �+4� �,&n�

�& . �+4� �,&+45,

dqg

s+n, . �� 5s+n,� @�&� �

&n� . ��+4� �,&

�& . �+4� �,&

@�+4� �,+�&3� . �+4� �,&3�,

�& . �+4� �,&=

+46,

Vxevwlwxwlqj htv1 +45, dqg +46, lqwr ht1 +44, |lhogv/ diwhu glylglqj e| gw/

ohwwlqj gw dssurdfk }hur dqg uhduudqjlqj whupv/ wkh Ehoopdq htxdwlrq=

+u. �,+�& . �+4� �,&,Y�+n, @�+�&n� . �+4� �,&n�,Y�+n . 4,

. ��+4� �,+�&3� . �+4� �,&3�,Y�+n � 4,=

+47,

Diwhu gh�qlqj I +n, =@ +�& . �+4� �,&,Y�+n,/ ht1 +47, fdq eh zulwwhq dv

+u . �,I +n, @ �I +n . 4, . ��+4� �,I +n � 4,= +48,

Ht1 +48, lv d vhfrqg rughu olqhdu krprjhqhrxv gl�huhqfh htxdwlrq zklfk kdv

dv jhqhudo vroxwlrq

I +n, @ D�&> +49,

zkhuh D lv d frqvwdqw dqg � lv d vroxwlrq ri wkh krprjhqhrxv htxdwlrq/

T+�, � �2�

u . �

�� . �+4� �, @ 3= +4:,

Ht1 +4:, kdv wzr uhdo urrwv/9 qdpho|

��c2 @u . �

5�

2

t+ o>. 4,2 � 7�+4� �,= +4;,

Qrwh wkdw T+3, @ �+4� �, A 3 dqg T+4 � �, @ �o>+4 � �, � 31 Vlqfh wkh

judsk ri T lv dq xszdug srlqwlqj sduderod zh pxvw kdyh �� � 4 � � dqg

3 ? �2 ? 4� � +vhh Iljxuh 4,1 Wkh ydoxh ixqfwlrq Y� lv wkhq jlyhq e|

8Wkh frqwlqxdwlrq uhjlrq lv wkh uhjlrq zkhuh lw lv rswlpdo wr uhiudlq iurp lqyhvwlqj dqg

wkxv zdlw iru pruh lqirupdwlrq19 Lw vkrxog eh qrwhg wkdw iru doo b lw krogv wkdw ebE� 3 b� $ �1 Vlqfh htxdolw| krogv l�

b ' �*2/ wkh krprjhqhrxv htxdwlrq lqghhg kdv wzr uhdo urrwv iru dq| b M E�*2c �o1

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Page 12: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

T+�,

3

4� �.

�2 ��

Iljxuh 4= Judsk ri T1

Y�+n, @I +n,

�& . �+4� �,&@

D��&� .D2�

&2

�& . �+4� �,&= +4<,

Wkh odvw fdvh lv wkh fdvh zkhuh n lv vxfk wkdw lw lv qrw rswlpdo wr lqyhvw

lq wkh surmhfw uljkw dzd|1 Krzhyhu/ li wkh iroorzlqj vljqdo lqglfdwhv kljk

uhyhqxhv/ lw zloo wkhq eh rswlpdo wr lqyhvw/ l1h nW�4 � n ? nW1 Lq wklv uhjlrq

wkh ydoxh ixqfwlrq Y2 iru wkh prqrsrolvw pxvw vdwlvi| wkh iroorzlqj Ehoopdq

htxdwlrq=

uY2 @�_|H+gY2,= +53,

Fdofxodwlqj ht1 +53, |lhogv/

uY2+n, @ �^s+n,+� +n . 4, . +4� �,Y�+n � 4,, . +4� s+n,,

+�Y�+n � 4, . +4� �, +n . 4,,� Y2+n,`

@ �^+5s+n,�. 4� �� s+n,, +n . 4, .

. +s+n, . �� 5s+n,�,Y�+n � 4,� Y2+n,`

/ +u . �,Y2+n, @ �^+5s+n,�. 4� �� s+n,, +n . 4, .

. +s+n, . �� 5s+n,�,Y�+n � 4,`=

+54,

Vxevwlwxwlqj htv1 +4<,/ +43,/ +45, dqg +46, lqwr ht1 +54, |lhogv

Y2+n, @�

u. �

��XMs+n,� +�s+n, . +4� �,+4� s+n,,,L

. �+4� �,��q

&3�

�n�2q

&3�

2

b&nlE�3b�&

�=

+55,

Li dq k0vljqdo duulyhv/ wkh surfhvv mxpsv wr wkh uhjlrq zkhuh n � nW dqg li

dq o0vljqdo dulyhv wkh surfhvv mxpsv wr wkh uhjlrq zkhuh n ? nW1 Wkhuhiruh

43

Page 13: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

wkh ydoxh Y2 lv frpsohwho| ghwhuplqhg e| Y�+n�4, dqg +n.4,1 Wkh ydoxh

ixqfwlrq Y lv wkhq jlyhq e| +fi1 htv1 +43,/ +4<, dqg +55,,

Y +n, @

;AAAAAAA?AAAAAAA=

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�n�2q

&

2

b&nlE�3b�&li n ? nW � 4

u . �

k�XMs+n,� +�s+n, . +4� �,

� +4� s+n,,,L . �+4� �,��q

&3�

�n�2q

&3�

2

b&nlE�3b�&

l li nW � 4 � n ? nW

XMs+n,� L li n � nW=

+56,

Wr ghwhuplqh wkh fulwlfdo ohyho nW/ zh xvh wkh iroorzlqj erxqgdu| frqglwlrqv1

Iluvw ri doo/ lw vkrxog krog wkdw li wkh qxpehu ri o0vljqdov uhodwlyh wr wkh

qxpehu ri k0vljqdov whqgv wr lq�qlw|/ wkhq wkh ydoxh iru wkh prqrsrolvw

vkrxog frqyhujh wr }hur/ l1h1

olp&<3"

Y +n, @ 3=

Wklv lpsolhv wkdw zh rqo| qhhg wr frqvlghu wkh odujhu urrw ��1: Khqfh/

D2 @ 31 Wr ghwhuplqh D� dqg nW zh vroyh wkh frqwlqxlw| frqglwlrq Y�+nW �

4, @ Y2+nW � 4, dqg wkh ydoxh0pdwfklqj frqglwlrq Y2+n

W, @ +nW,1 Vroylqj

wkh odwwhu htxdwlrq iru D� |lhogv

D� @4

�&W

3�� ��+4� �,

^XM�&W

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+57,

Vxevwlwxwlqj D� lq wkh iruphu htxdwlrq |lhogv dq h{suhvvlrq iru sW � s+nW,=

sW @4

�+XM@L � 4, . 4> +58,

zkhuh

� @��+u. �,+u. �+4� �,,� ��+4� �,+u . �+4 . �� � �,,

��+u . �,+u. ��,� ��+4� �,+u . �+�� . �,,=

Wkh rswlpdo qxpehu ri k0vljqdov uhodwlyh wr o0vljqdov lv wkhq jlyhq e|

nW @orj+ RW

�3RW, . orj+�,

orj+ b�3b

,= +59,

:Wklv vwhpv iurp wkh idfw wkdw q2 �3b/ vr lq T�E&� dqg T2E&� wkh whup q&2 grplqdwhv

li & < 3" frpsduhg zlwk E�3 b�&1 Khqfh/ li �2 �' f/ wkhq T E&�<" li & < 3"1

44

Page 14: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

Iurp ht1 +59, lw lv rewdlqhg wkdw nW ghfuhdvhv zlwk sf1 Khqfh/ ohvv dgglwlrqdo

lqirupdwlrq lv qhhghg zkhq wkh lqlwldo eholhi lq kljk uhyhqxhv lv douhdg| kljk1

Qh{w/ zh fkhfn zkhwkhu wkh rswlpdo eholhi sW lv d zhoo0gh�qhg suredelolw|1

Wkh iroorzlqj sursrvlwlrq hvwdeolvkhv wklv uhvxow/ zklfk lv suryhg lq wkh

dsshqgl{1 Lw ixuwkhupruh vkrzv wkh olqn ehwzhhq wkh rswlpdo dssurdfk dqg

wkh wudglwlrqdo QSY uxoh1

Sursrvlwlrq 614 Iru +XM,W � L lw krogv wkdw sW � 41 Ixuwkhupruh/ sW A

s��T 1

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rswlpdo wr zdlw vrphzkdw orqjhu iru pruh lqirupdwlrq wr dulyh wkdw uhgxfhv

xqfhuwdlqw|1

Iru frqyhqlhqfh zh vxppdul}h wkh uhvxow ri wkh rswlpdo dssurdfk lq wkh

iroorzlqj wkhruhp1

Wkhruhp 614 Wkh vhw ri frpelqdwlrqv ri qxpehu ri vljqdov +q, dqg k0

vljqdov +j,/ iru zklfk lw lv rswlpdo wr lqyhvw lv jlyhq e|

E =@ i+q> j, 5 LQ� LQmj � q> 5j � q � enWf. 4j= +5:,

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vwdwhg lq wkh iroorzlqj sursrvlwlrq/ wkh surri ri zklfk lv jlyhq lq wkh ds0

shqgl{1

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dqg � dqg ghfuhdvhv zlwk XM 1 Lw lqfuhdvhv zlwk � l�

t+ o>. 4,2 � 7�+4� �, �

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= +5;,

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wlrq krogv hyhu|zkhuh/ h{fhsw zkhq o2

>2lv forvh wr 5� � 4/ l1h1 li u �

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Wkh ukv ri wkh dssur{lpdwlrq fdq eh vhhq dv d phdvxuh ri wkh xvhixoqhvv ri

lqirupdwlrq lq d xqlw ri wlph/ vlqfh lw lv wkh h{shfwhg qxpehu ri vljqdov shu

wlph xqlw wlphv wkh suredelolw| ri d fruuhfw vljqdo plqxv wkh suredelolw| ri

dq lqfruuhfw vljqdo1 Vr/ wkh frqglwlrq lv qrw vdwlv�hg li wkh glvfrxqw udwh lv

dssur{lpdwho| wkh vdph dv wkh lqirupdwlyhqhvv ri wkh vljqdov1

45

Page 15: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

Dq lpsruwdqw txhvwlrq wkh prqrsrolvw idfhv lv krz olnho| lw lv wkdw kh

pdnhv d zurqj ghflvlrq/ lq wkh vhqvh wkdw kh lqyhvwv zkloh wkh pdunhw lv edg1

Wklv txhvwlrq fdq eh dqvzhuhg txdqwlwdwlyho| e| fdofxodwlqj wkh suredelolw|

wkdw nW lv uhdfkhg zkloh uhyhqxhv duh orz1 Lq rughu wr gr vr/ gh�qh

S E&� =@ Su+inW lv uhdfkhg vwduwlqj iurp n/

jlyhq wkdw uhyhqxhv duh orzj,=+5<,

D vhfrqg rughu olqhdu gl�huhqfh htxdwlrq fdq qrz eh rewdlqhg jryhuqlqj

S E&�=

S E&� @ +4� �,S E&n�� . �S E&3��= +63,

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S E&W� @ 4

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olp&<3"

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rqh fdq vroyh ht1 +63,1 Wklv |lhogv

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vljqdov lqfuhdvhv1 Wkh h{ dqwh suredelolw| ri d zurqj ghflvlrq lv jlyhq e|

S E&f�1

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d orvv1 Wkhuhiruh/ wkh prqrsrolvw zloo qhyhu lqyhvw/ hyhq li kh eholhyhv wkdw

pdunhw ghpdqg lv kljk iru vxuh1

46

Page 16: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

Wkh pruh lqwhuhvwlqj fdvh lv wkh vhfrqg rqh1 Qrz wkhuh lv dq rswlrq

ydoxh ri zdlwlqj iru pruh lqirupdwlrq1 Ohw xv frqvlghu dq h{dpsoh zkhuh

d prqrsrolvw idfhv wkh sureohp zkhwkhu ru qrw wr lqwurgxfh d qhz surgxfw

wr wkh pdunhw1 Pdunhw ghpdqg fdq eh hlwkhu srvlwlyh li wkh pdunhw lv jrrg

ru }hur li wkh pdunhw lv edg1 Lq wkh vwdqgdug plfurhfrqrplf dqg Lqgxvwuldo

Rujdql}dwlrq olwhudwxuh wklv lv qrw frqvlghuhg wr eh d sureohp/ vlqfh wkh QSY

dssurdfk lv xvhg1 Khqfh/ wkh rswlrq ydoxh ri zdlwlqj iru pruh lqirupdwlrq

lv qrw lqfoxghg1 Wkh vwudwhjlf dvshfwv ri lqyhvwlqj duh ghdow zlwk txlwh

h{whqvlyho| +fi1 Pdv0Frohoo hw do1 +4<<8, dqg Wluroh +4<;;,,1 Khuh wrr/

rswlrq ydoxhv duh ljqruhg1 Lq pruh uhfhqw frqwulexwlrqv wklv rswlrq ydoxh lv

lqfoxghg lq wkh dqdo|vlv ri vwudwhjlf wlplqj ri lqyhvwphqw +fi1 Krssh +5333,,

dqg Kxlvpdq +5333,,1

Lqyhuvh ghpdqg lv dvvxphg wr eh jlyhq e| wkh iroorzlqj ixqfwlrq/

S +t, @

;?=\ � t li t � \ dqg K

3 rwkhuzlvh>+65,

zkhuh t lv wkh txdqwlw| vxssolhg1 Wkh frvwv ri surgxflqj t xqlwv duh jlyhq

e| wkh frvw ixqfwlrq

F+t, @ ft> f � 3= +66,

Wkh sur�w ri surgxflqj t xqlwv lv wkhq jlyhq e|

�+t, @ S +t,t �F+t,= +67,

Vxssrvh iru d prphqw wkdw wkh pdunhw pdunhw lv jrrg1 Wkhq wkh uhyhqxh

wr wkh prqrsrolvw lv jlyhq e|/

U} @ pd{^

�]"

f

h3o|�+t,gw

@ pd{^i�+t,�

oj=

+68,

Vroylqj iru t xvlqj wkh �uvw rughu frqglwlrq |lhogv wkh rswlpdo rxwsxw ohyho

tW @ t3S2/ ohdglqj wr wkh rswlpdo uhyhqxh

XM @ �o^S +tW,tW �F+tW,`= +69,

Li wkh pdunhw lv edg lw lv rswlpdo qrw wr surgxfh dw doo1 Khqfh/ wkh uhyhqxh

li ghpdqg lv }hur/ Xu/ lv jlyhq e|/

Xu @ 3= +6:,

47

Page 17: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

\ @ ; u @ 3=4

f @ 8 � @ 7

L @ 45 � @ 3=;

sf @�2

Wdeoh 5= Sdudphwhu ydoxhv

Lq wklv h{dpsoh zh dvvxph wkdw wkh prqrsrolvw kdv qr d sulrul uhdvrq wr

vxvshfw wkh pdunhw wr eh jrrg ru edg1 Khqfh/ lw krogv wkdw sf @�21

Wkh fkrvhq sdudphwhu ydoxhv duh olvwhg lq Wdeoh 51 Vr/ wkh glvfrxqw udwh

u lv vhw dw 43(1 Lqwhusuhwlqj u duh dv wkh ulvniuhh lqwhuhvw udwh/ wklv lv d

frpprq ydoxh lq wkh olwhudwxuh1 Wkh suredelolw| ri d fruuhfw vljqdo lv 31;

dqg rq dyhudjh irxu vljqdov duulyh hyhu| |hdu1

Edvhg rq wkhvh sdudphwhu ydoxhv wkh ydoxh ixqfwlrq lv fdofxodwhg dv ixqf0

wlrq ri n dqg ghslfwhg lq Iljxuh 51 Iurp wklv �jxuh rqh fdq vhh wkdw wkh

QSY uxoh suhvfulehv qrw wr lqyhvw dw wkh prphqw wkh rswlrq ehfrphv dydlo0

deoh +n @ 3,1 Lq idfw/ lq rughu wr lqyhvw/ wkh QSY uxoh ghpdqgv wkdw wkh

eholhi ri wkh prqrsrolvw lq kljk pdunhw ghpdqg vkrxog dw ohdvw eh dssur{0

lpdwho| 3186 +n��T � 3=43,1 Krzhyhu/ wkh rswlpdo dssurdfk vshfl�hv wkdw

wkh prqrsrolvw*v eholhi vkrxog h{fhhg sW � 3=<9 +nW � 5=56,1 Wklv lpsolhv

wkdw iurp ht1 +<, lw fdq eh ghulyhg wkdw wkuhh k0vljqdov vkrxog duulyh ehiruh

lw lv rswlpdo wr lqyhvw/ surylghg wkdw qr o0vljqdov rffxu1 Khqfh/ wkh prqrs0

rolvw wkhq vkrxog rq dyhudjh zdlw wkuhh txduwhuv ri d |hdu ehiruh hqwhulqj

wkh pdunhw1 Wkh QSY uxoh suhvfulehv wkdw/ lq devhqfh ri o0vljqdov/ rqo| rqh

k0vljqdo lv qhhghg1 Vr/ wkh rswlpdo dssurdfk suhvfulehv wr zdlw orqjhu wkdq

wkh QSY uxoh1 Wkh gl�huhqfh pd| eh kxjh1 Ixuwkhupruh/ iurp ht1 +64, lw lv

rewdlqhg wkdw wkh suredelolw| ri wdnlqj d zurqj ghflvlrq xvlqj wkh rswlpdo

dssurdfk htxdov S Ef� @ 3=334891 Xvlqj wkh QSY uxoh lpsolhv d suredelolw|

ri 31581 Khqfh/ wkh suredelolw| ri pdnlqj d zurqj ghflvlrq xvlqj wkh rswlpdo

dssurdfk lv pxfk orzhu wkdq zkhq wkh QSY uxoh lv xvhg1

Xvlqj wkh vdph sdudphwhuv zh fdq vhh krz wkh fulwlfdo ydoxh nW fkdqjhv

zlwk ydoxhv iru u/ �/ dqg �1; Wkh fdvh zkhq u lv ydulhg lv ghslfwhg lq Iljxuh 61

Lw vkrzv wkdw li wkh ulvniuhh udwh lqfuhdvhv/ wkh fulwlfdo ohyho ulvhv1 Wklv lv

wkh vr0fdoohg suhvhqw ydoxh h�hfw1 Li u lqfuhdvhv ixwxuh lqfrph lv ydoxhg ohvv

;Rqh fdq vhh wkdw R

W

lv qrw lq xhqfhg e| Rf/ dowkrxjk &W

lv1 Zh douhdg| vdz wkdw wkh

odwwhu txdqwlw| ghfuhvhv lq Rf1

48

Page 18: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

-2 -1 0 1 2 3 4-15

-10

-5

0

5

10

15

k

valu

e

k*

Iljxuh 5= Ydoxh ixqfwlrq1 Wkh gdvkhg olqh ghqrwhv wkh QSY1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.21.5

2

2.5

3

3.5

4

4.5

5

5.5

r

k*

Iljxuh 6= Frpsdudwlyh vwdwlfv iru u

vr wkdw wkh qhw suhvhqw ydoxh ghfuhdvhv1 Wkhuhiruh/ wkh prqrsrolvw lv zloolqj

wr zdlw orqjhu zlwk lqyhvwphqw xqwlo kh kdv pruh fhuwdlqw| derxw wkh dfwxdo

vlwxdwlrq lq wkh pdunhw1 Wkh dv|pswrwlf ehkdylrxu rffxuv ehfdxvh ri wkh

erxqg rq u wkdw lv lpsolflwo| jlyhq e| htv1 +69, dqg +;,1

Lq Iljxuh 7 zh fdq vhh wkdw nW lqfuhdvhv zlwk �1< Wklv lv ehfdxvh li vljqdov

duulyh pruh iuhtxhqwo|/ lw wdnhv ohvv wlph wr jhw lqirupdwlrq rq wkh dfwxdo

pdunhw frqglwlrqv1 Wkhuhiruh/ wkh prqrsrolvw zloo zdlw iru pruh vljqdov wr

duulyh1 Li � whqgv wr lq�qlw|/ l1h1 li wkh lqwhuduulydo wlph ehwzhhq vljqdov

frqyhujhv wr }hur/ wkhq wkh fulwlfdo ydoxh sW whqgv wr rqh/ vlqfh zlwk vr pdq|

<Qrwh wkdw ht1 +5;, lv vdwlv�hg1

49

Page 19: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

µ

k*

Iljxuh 7= Frpsdudwlyh vwdwlfv iru �

vljqdov duulylqj lpphgldwho|/ wkh xqfhuwdlqw| glvdsshduv lpphgldwho|143

Ilqdoo|/ iurp Iljxuh 8 zh fdq frqfoxgh wkdw wkh fulwlfdo ohyho iru wkh

eholhyh lq kljk uhyhqxhv lqfuhdvhv zlwk wkh txdolw| ri wkh vljqdo �1 Li � lv

0.5 0.6 0.7 0.8 0.9 10.5

0.55

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0.65

0.7

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0.85

0.9

0.95

1

λ

p*

0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

λ

k*

Iljxuh 8= Frpsdudwlyh vwdwlfv iru �

kljkhu/ wkhq wkh lqirupdwlyhqhvv ri d vljqdo lv kljkhu1 Vr/ lw lv pruh dwwudfwlyh

iru wkh prqrsrolvw wr ghpdqg d kljkhu fhuwdlqw| derxw wkh jrrgqhvv ri wkh

43Wklv fdq eh vkrzq irupdoo| e| qrwlqj wkdw

?E[� _E[� '>2b2Eq� 3 E�3 b�� n >Eq�o n

n obEq� 3 E� 3 b��� n q�o2<" li ><"c

zkhuh ?E[� lv wkh qxphudwru dqg _E[� lv wkh ghqrplqdwru ri [1 Khqfh/

*�4><"

[ ' fc +6;,

zklfk lpsolhv wkdw *�4><"

RW ' �1

4:

Page 20: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

pdunhw1 Wklv eholhi krzhyhu/ lv uhdfkhg diwhu ihzhu vljqdov dv fdq eh vhhq

iurp wkh uljkw sodqh ri Iljxuh 81

! "��������

Lq wklv sdshu d vlwxdwlrq zdv dqdo|vhg zkhuh d prqrsrolvw kdv wkh rssru0

wxqlw| wr hqwhu d qhz pdunhw1 Pdunhw frqglwlrqv duh xqfhuwdlq/ exw dv wlph

sdvvhv wkh prqrsrolvw uhfhlyhv vljqdov derxw wkh sur�wdelolw| ri wkh lqyhvw0

phqw1 Lw zdv vkrzq wkdw wkh wudglwlrqdo qhw suhvhqw ydoxh uxoh idlov wr uh hfw

wkh rswlrq ydoxh wkdw lv fuhdwhg e| wkh rssruwxqlw| wr zdlw iru pruh vljqdov

wr duulyh1 Wkh suhvhqw sdshu gl�huv iurp wkh vwdqgdug olwhudwxuh rq lqyhvw0

phqw xqghu xqfhuwdlqw| +vhh Gl{lw dqg Slqg|fn +4<<7,, lq wkdw xqfhuwdlqw|

uhvroyhv lq wkh frxuvh ri wlph/ zklfk vhhpv d uhdolvwlf dvvxpswlrq lq hyhu|

gd| olih1

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wr hqwhu wkh pdunhw1 Wklv zrxog uhtxluh xvlqj jdph wkhruhwlf frqfhswv lq

wkh suhvhqw vhwwlqj1 Dqrwkhu wrslf iru ixuwkhu uhvhdufk pljkw eh wr lqfoxgh

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h{whqg wkh lghd ri uhvroylqj xqfhuwdlqw|1 Iru lqvwdqfh/ wr orrn dw d pdunhw

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frpshwlwlrq zrunv lq dq xqfhuwdlq hqylurqphqw/ zklfk frxog iru lqvwdqfh

eh dssolhg wr dqwlwuxvw sureohpv1 Vlqfh wkh dssurdfk lq Gl{lw dqg Slqg|fn

+4<<7, grhv qrw wdnh lqwr dffrxqw ydqlvklqj xqfhuwdlqw|/ rqh frxog vxvshfw

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frxog ohdg wr d odlvvh}0idluh srolf| wkdw lv wdnhq wrr idu/ vlqfh rqh zrxog

ryhuhvwlpdwh h1j1 wkh sulfh pdujlq wkdw vkrxog eh doorzhg gxh wr xqfhuwdlqw|1

Xvlqj wkh QSY dssurdfk zrxog ohdg wr dq ryhu0dfwlyh dqwlwuxvw srolf|/ dv lv

uljkwo| srlqwhg rxw lq Gl{lw dqg Slqg|fn +4<<7/ fk1 <,/ vlqfh wklv dssurdfk

grhv qrw wdnh lqwr dffrxqw dq| uh hfwlrq ri xqfhuwdlqw| lq wkh sulfh pdujlq1

Dsso|lqj wkh lghdv suhvhqwhg lq wklv sdshu wr dq hvwdeolvkhg pdunhw dqdo|vlv

zh zrxog h{shfw wkdw wkh pdun0xs ri xqfhuwdlqw| lq wkh sulfh pdujlq lq d

4;

Page 21: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

fhuwdlq lqgxvwu| ghshqgv fulwlfdoo| rq wkh ohyho ri pdwxulw| lq wkh lqgxvwu|1

� ����# �# ��������� ���

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dwru ri � e| q+�,1 Qrwh wkdw +xvlqj �� � 4� �,

q+�, @ ��+u . �,+u. ��,� ��+4� �,+u . �+�� . �,,

� �+5�� 4,^��+u . �,� ��+4� �,`

� ��+u . �,+u. ��,� ��+4� �,+u . �+�� . �,,

� �+5�� 4,^u+4� �, . �+4� �,2`

? g+�,=

+6<,

Khqfh/ � ? 41 Li u @ 3/ lw krogv wkdw �� @ �1 Wkhuhiruh/

q+3, @ ��2+4� �,� ��+4� �,� @ 3=

Ixuwkhupruh/ xvlqj wkdw �� � +4� �,/ lw fdq eh rewdlqhg wkdw

gq+�,

gu@ 5��u� ��+4� �, . ���+5� �,

.C��

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2+4� �,2�

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Cu

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2+4� �,2�

A 3>

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A 31 Vr/ � A 3 dqg sW lv d zhoo0gh�qhg suredelolw|1 Ixuwkhupruh/

vlqfh � ? 4/ lw krogv wkdw

sW @

4

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4<

Page 22: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

$ ����# �# ��������� ���

Vlpsoh fdofxoxv jlyhv wkh uhvxow iru XM dqg L1 Wr suryh wkh sursrvlwlrq

iru u/ �/ dqg �/ ohw xv �uvw ghulyh wkh frpsdudwlyh vwdwlfv ri �� iru wkhvh

sdudphwhuv1 Iluvw/ wdnh u1 Wkh wrwdo gl�huhqwldo ri T zlwk uhvshfw wr u lv

jlyhq e|

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A 31 Ixuwkhupruh/ Y%Yo

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ghqrplqdwru ri � fdq eh zulwwhq lq wkh iroorzlqj irup

q+�, @ �+u> �> �,� 5�+4� �,�+u> �> �,>

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Vlqfh � A 3/ wklv lpsolhv wkdw wr ghwhuplqh wkh vljq ri wkh ghulydwlyh ri

� zlwk uhvshfw wr rqh ri wkh sdudphwhuv/ rqh rqo| qhhgv wr frpsduh wkh

uhvshfwlyh ghulydwlyhv ri �+�, dqg �+�,1 Qrwh wkdw

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Cu=

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Khqfh/ Y[Yo

? 3 dqg YRW

YoA 31

53

Page 23: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

Iru � d vlplodu h{huflvh fdq eh grqh/ |lhoglqj

C�+�,

C�@ �+5�� 4, . +u . �,

C��

C�A 3

A ��u . 5�+4� �,

�@

C�+�,

C�=

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YbA 31

Wr suryh wkh uhvxow rq �/ rqh qhhgv wr fdofxodwh Yq�Y>

h{solflwo|1 Wklv

|lhogv

C��

C�@ �u

��

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= +76,

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C�+�,

C�@

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/

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Wkh odvw lqhtxdolw| lv suhflvho| ht1 +5;,1 Vr/ wkhq YRW

Y>A 31 �

54

Page 24: Tilburg University Investment Under Vanishing Uncertainty …No. 2001-14 INVESTMENT UNDER VANISHING UNCERTAINTY DUE TO INFORMATION ARRIVING OVER TIME By Jacco J.J. Thijssen, Eric E.C

%�#������

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55